How this instrument works
Absolute uncertainty expresses how far a reported value might reasonably be from the true one, in the same units as the measurement itself — 24.4 ± 0.2 cm means the true length is believed to sit somewhere within 0.2 cm of 24.4 cm. There is more than one accepted way to estimate that uncertainty from a set of repeated readings; this instrument uses the half-range method, a common introductory-lab convention: take the best estimate as the mean of the readings, and the uncertainty as half the distance between the largest and smallest reading.
The logic is straightforward. If five repeated measurements of the same quantity scatter across a certain spread, that scatter itself is the clearest evidence of how precise the measurement process is — a tight cluster implies a small uncertainty, a wide scatter implies a large one. Taking half the full range gives a single number that describes how far a typical reading might sit from the center, without requiring any more advanced statistical machinery.
This is a simpler, more approximate technique than propagating a full standard deviation or standard error of the mean, and it is more sensitive to a single outlying reading than those methods are, since it depends only on the two most extreme values rather than every value. It is well suited to the small repeat counts common in a teaching lab (three to six readings); for larger data sets or more rigorous work, a standard deviation or standard-error-based uncertainty, or the fuller framework in the international GUM, is the more defensible choice.
- Enter your repeated measurements of the same quantity into Repeated measurements, separated by commas, spaces, or new lines.
- Enter at least 2 values — more repeats generally give a steadier best estimate, though the half-range method always uses only the extremes for the uncertainty itself.
- Read Best estimate (mean) for the reported central value.
- Read Absolute uncertainty (± half the range) for the plus-or-minus figure to report alongside it.
Worked example — five lab thermometer readings
A student reads a lab thermometer five times over a few minutes while the room is meant to be at a steady temperature: 24.2°C, 24.5°C, 24.3°C, 24.6°C, and 24.4°C. Entering those five values, Best estimate (mean) reads 24.4, computed as (24.2+24.5+24.3+24.6+24.4)/5 = 122.0/5 = 24.4 exactly.
Absolute uncertainty (± half the range) reads 0.2, computed as (24.6 − 24.2)/2 = 0.4/2 = 0.2 — the highest and lowest of the five readings are 0.4°C apart, so half that spread is reported as the uncertainty. The result would be written up as 24.4 ± 0.2°C, meaning the true room temperature is estimated to lie within about 0.2°C of 24.4°C based on this set of readings.
Questions
Why divide the range by 2 instead of using the standard deviation?
The half-range method is a deliberately simple, low-overhead estimate meant for small repeat counts, like the three-to-six readings typical in an introductory lab — it needs only the two extreme values and no further arithmetic. A standard deviation (or the standard error of the mean, dividing further by √n) uses every reading and is the more statistically rigorous choice, especially once you have more than a handful of measurements; it just takes more steps to compute by hand.
Is absolute uncertainty the same as a standard error?
No. The half-range method reported here depends only on the largest and smallest of your readings, so a single unusually high or low reading changes it directly. A standard error of the mean instead reflects the typical spread of every reading, divided by the square root of the sample size, which makes it steadier as you add more measurements — the two numbers can differ noticeably on the same data set.
How many repeated measurements should I take?
At least 2 are required to have any range to work with, but 3 to 6 repeats is the typical range for this method in a teaching lab. More repeats give the mean a steadier footing, though because the uncertainty itself uses only the maximum and minimum, adding more readings mainly changes it if one of the new readings sets a new extreme.
What if all my measurements come out identical?
Then the uncertainty comes out to exactly zero, since max and min are the same value — this reflects a genuine lack of observed variation across your specific readings, not a claim that the true value is known with perfect precision. Instrument resolution and other error sources not captured by repeat-to-repeat scatter can still leave the true value slightly uncertain even when every reading matches.
Does the uncertainty carry the same units as the measurements?
Yes — absolute uncertainty is always expressed in the same units as the quantity itself (seconds, grams, degrees, and so on), which is what distinguishes it from a relative or percent uncertainty. If you need the relative version, divide this instrument's uncertainty by its best estimate and multiply by 100 to get a percentage.